unit 1 - trees

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Trees

MPSTME, SHIRPUR

Preeti S. Patil 1

TREES

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Trees

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Preeti S. Patil 2

Introduction to Trees Flexible, versatile, and powerful data

structures.

Used to represent hierarchical relationship

among them. Example: Geneology

The relationship between the grandfather and

his descendants and in turn their descendantsand so on

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Tree definition Non-linear data structure in which items

are arranged in a sorted sequence.

Represents hierarchical relationshipexisting amongst several data items.

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Graph theoretic definitionIt is finite set of one or more data items

(nodes) such that

1. There is special data item called rootof the

tree2. Remaining data items are partitioned into

number of mutually exclusive (i.e, disjoint)

subsets. Each of which is itself istree called to

be subtrees

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A Tree

A

B C D

E F G H I

J K L M

Level 0

Level 1

Level 2

Level 3

Root

Trees in data structures grows from top to bottom.

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Tree TerminologyTerm associated with trees1. Root

2. Node

3. Degree of a node

4. Degree of a tree

5. Terminal node(s)6. Non-terminal node(s)

7. Siblings

8. Level

9. Edge10. Path

11. Depth

12. Forest

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Root

1. Specially designated data item.

2. First in the hierarchical arrangements ofdata items.

Node

1. Each data item in a tree is called tree.2. It specifies the data information and links

(branches) to other data items.

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Degree of a node

It is the number of sub trees of a node in agiven tree.

a. The degree of nodeA is 3

b. The degree of node C is 1

c. The degree of node D is 2

d. The degree of node H is 3

e. The degree of node I is 0

Degree of treeIt is maximum degree of the nodes in agiven tree.

In the tree shown , the node A has degree 3 and another nodeH is also having its degree 3. in all the value is the maximum.So, the degree of the tree is 3

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Terminal Node(s)

A node with degree zero.

Also called leaf node. Ex. Nodes E,G,I,J,K,L and M are Terminal nodes

Non-Terminal Node(s)

Any node (except the root node) whose degree is not

zero.

Intermediate nodes in traversing the given tree from

its root to the terminal nodes.

Ex nodes B, C, D, F & H are Non terminal nodes

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Siblings

The children nodes of a given node.

Also called brothers.

E and F are siblings of parent node B.

K, L and M are siblings of parent node H.

Level Entire tree structure is leveled as such that the root is

always at level 0.

Its intermediate at level 1. and their intermediate atlevel 2 and so on upto the terminal nodes.

Tree shown has level 4.

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Edge

Connecting line of two nodes. line drawn from one node to another node.

Path

Sequence of consecutive edges from source

node to the destination node.

Eg: Path betweenA and J is given by

(A,B), (B,F), (F,J)

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Depth/height

Maximum level of any node in a given tree. Number of levels one can descend the tree from its

root to the terminal nodes

In tree shown Root node has the maximum level. Forest

Set of disjoint trees.

Ifroot is removed from tree it becomes forest. In tree shown there is forest with three trees

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Binary Trees Finite set of data items which is either

empty or consists of a single item calledroot and two disjoint binary trees called left

sub tree and right sub tree

Maximum degree of any node is at most

two.

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A binary tree

A

B C

D E F G

J

Root

I K

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Complete Binary Tree There is exactly one node at level 0,two

nodes at level 1, four nodes at level 2 andso on.

If there are m nodes at level L then a

binary tree contains 2m nodes at level

L+1.

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Properties of binary trees

1. The maximum number of nodes at level l of a binary

tree is 2l, where l>=0

2. The maximum number of nodes in a binary tree of

depth d is 2d-1 where d>=1.

3. The total number of edges in a complete binary tree

with n terminal nodes is 2(n-1).

4. A binary tree with n nodes has exactly N+1 null

branches.

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TreesMPSTME, SHIRPUR

Preeti S. Patil 18

Binary Tree Representation Arrays representation

Linked list representation

1. Basic component of a tree is a node2. Node consist of three fields.

Data

Left Child

Right Child

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Data field hold the value to be given. The left child is a link field which contains

the address of its left node.

The right child is a link field which contains

the address of its right node.

The Structure of a node

Lchild Data Rchild

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Logical Representationstruct node

{

char data;

struct node lchild;struct node rchild;

};typedef struct node BTNODE;

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Consider the following Binary tree.

A

B C

D E

Root

F

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Linked List representation

A

B C

D E F

Root

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Array Representation The nodes stored in an array are

accessible sequentially. Maximum nu ber of nodes is specified by

MaxSize.

Root node is always at index 0.

Successive memory locations the left child

and right child are stored.

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The array representation of the above tree

A

B C

0

1 2

A

B

C

BT

BT [0]

BT [1]

BT [2]

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A

B C

D E F G

0

2

5 6

1

3 4

A

B

C

DE

F

G

BT

BT [0]

BT [1]

BT [2]

BT [3]BT [4]

BT [5]

BT [6]

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Father(n):

The Father of node having index n is at floor((n-1)/2) If n=0, then it is the root node and has no father.

Eg: father of node numbered 3(i.e, D)

=floor ((3-1) / 2)=floor (2/2)

=1(i.e B)

lchild(n): The left child of node numbered n is at (2n+1).

Eg: lchild(A)=lchild(0)

=2*0+1

=1(i.e B).

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rchild(n):

The right child of node numbered n is atindex(2n+2)

Eg: rchild(A)=rchild(0)

=2*0+2=2(i.e C).

Siblings: If left child at index n is given then right sibling

is at (n+1).

If right child at index n is given then left siblingis at (n-1).

Right node of node 4 is at index 5.

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Skewed binary tree

Since left sub tree is present, this type of tree is called left skewedbinary tree. we can even have right skewed binary tree. This result

in wastage of memory in case of array representation.

A

B

C

0

1

3

A

B

-

C

-

-

-

BT

0

1

2

3

4

5

6

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Disadvantages of array

representation

Not suitable for normal binary trees.

Ideal for complete binary trees

More memory is unnecessarily wasted.

Insertion and deletion at the middle of thetree is cumbersome.

O ti bi t

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Operations on binary trees

Operation Description1. Create It creates an empty binary tree.

2. Make BT It creates a new Binary Tree having single node with data

field set to some value.

3. Empty BT It returns true if the binary tree is empty. Otherwise it returns

false.

4. lchild It returns a pointer to the left child of the node. if the node has

no left child it returns the null pointer

5. rchild It returns a pointer to the right child of the node. if the node

has no right child it returns the null pointer

6. father It returns the pointer to father of the node.otherwise returns

null pointer

7. Brother It returns the pointer to brother of the node.otherwise returns

null pointer

8. Data It returns the content of the node.

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Operations Other than primitive operation other

operations are1. Tree traversal

2. Insertion of nodes

3. Deletion of nodes

4. Searching for the node

5. Copying the binary tree.

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Traversal of a Binary Tree It is a way in which each node in the tree

is visited exactly once in systematicmanner.

Popular ways of Binary tree traversal1. Preorder Traversal

2. Postorder Traversal

3. Inorder Traversal

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Preorder TraversalThe preorder of a non-empty binary tree is

defined as

1. Visit the root node.2. Traverse the left sub tree in preorder.

3. Traverse the right sub tree in preorder.

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Preorder traversal of the above binary tree

A B D E I C F G JRoot

A

B C

D E F G

J

Root

I

left Right

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Algorithmpreorder(BTNODE *tree)

{if(tree != NULL)

{

printf(%c,tree->data); /* step 1*/preorder(tree->lchild); /*step 2*/

preorder(tree->rchild); /*step 3*/

}return;

}

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Inorder TraversalThe Inorder of a non-empty binary tree is

defined as

1. Traverse the left sub tree in inorder.2. Visit the root node.

3. Traverse the right sub tree in inorder.

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A

B C

D E F G

J

Root

I

left Right

Inorder traversal of the above binary treeD B I E A F C G J

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Algorithminorder(BTNODE *tree)

{if(tree != NULL)

{

inorder(tree->lchild); /*step 1*/printf(%c,tree->data); /* step 2*/

inorder(tree->rchild); /*step 3*/

}

return;

}

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Postorder TraversalThe postorder of a non-empty binary tree is

defined as

1. Traverse the left sub tree in postorder.2. Traverse the right sub tree in postorder.

3. Visit the root node.

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Post order traversal of the above binary tree

D I E B F J G C A

A

B C

D E F G

J

Root

I

left Right

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Algorithmpostorder(BTNODE *tree)

{if(tree != NULL)

{

postorder(tree->lchild); /*step 1*/

postorder(tree->rchild); /*step 2*/

printf(%c,tree->data); /* step 3*/

}

return;

}

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Binary Search Tree (BST)

Tree in which each node has left and rightchild.

If traversed in inorder, we get a file is

ascending order.

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Traverse the tree in inorder, we get

9 12 21 25 42 43 48

25

12 43

9 21 42 48

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SEARCHING Start with the root.

Eg; search 21 in the given tree Compare 2112. so take right branch.

On third comparison, since 21=21, the search

is announced successful.

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Advantages Advantage over an array is that all the

insertion and deletions and searching areperformed efficiently.

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Height Balanced Tree We attempt to place each terminal node at

equal distance from the root node.

A

B C

D E F G

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Insertion in a BST To insert an element 23 in an tree shown below

Search for element 23 in that tree.

Search causes to reach at the node 21 with no where to go.

Since 23>21, we simply insert node 23 as right child to node 21.

25

12 43

9 21 42 48

25

12 43

9 21 42 48

23

23>21

Algorithm for SearchingS h(i t k l i t &f d BTNODE *(& t))

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Search(int key_val, int &found, BTNODE *(&parent))

{

BTNODE *p;

found=0;

parent=NULL;

if(tree==NULL)

{

return;

}

p=tree;while(p!=null)

{

if(p->data==key_val)

{found=1;

return;

}if(p->data > key_val)

{parent=p;

p=p->lchild;

}

else

if(p->data < key_val){parent=p;

p=p->rchild;

}

}}

Algorithm for insertion

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ginsert BST(int num)

{int found;

BTNODE *temp,*parent;

search_node(num,found,parent);

if(found==1)

printf(Node already exists);else

{

temp=(BTNODE*)malloc(sizeof(BTNODE));

temp->data=num;

temp->lchild=temp->rchild=NULL;if(parent==NULL)

tree=temp;

else

{ if(parent->data > num)

parent->lchild=temp;else

parent-> rchild=temp;

}}}

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Deletion in a BST

Three cases:

1. Node to be deleted has no children.

2. Node to be deleted has exactly one

subtree.

3. Node to be deleted has two subtrees.

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In first situation ,

As there is no children for the node,so that node is deleted with out any

further adjustments to the tree.

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In second case,

The node to be deleted has exactly one child, so this child is

moved up the tree to occupy its place.25

12 43

9 21 42 48

23Node to

be

deleted

25

12 43

9 23 42 48

After Deletion of the node 21

Before Deletion of the node 21

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Third case,The node to be deleted has two sub trees. The in order

successor of the node should be moved up to occupy its

place.

25

12 43

9 21 42 48

23

Node to

be

deleted

42

12 43

9 21 48

23

After Deletion of

the Root node 25

Before Deletion of

the Root node 25

C code for deletiond l BST(i b BTNODE * )

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delete_BST(int number, BTNODE *tree)

{int found;

BTNODE *temp, *parent, *temp_succ;

if(tree==NULL)

{

printf( Tree is empty);

return;

}

parent=temp=Null;

search_BST(number,parent,temp,tree,found)

if(found==0)

{ printf( node to be deleted is not found);return;

}

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/* first case : The node to be deleted has no children*/If((temp->lchild == NULL) && (temp->rchild ==NULL))

{

if(parent-.rchild == temp){

parent->rchild = NULL;

}

else{

parent->lchild= NULL;

}

free(temp);return;

}

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/* second case: if node to be deleted has only left child */If(temp->lchild != NULL)&&(temp->rchild == NULL)

{

if(parent->lchild ==temp)

{parent->lchild =temp->lchild;

}

else

{

parent->rchild = temp->lchild;

}

free(temp);return;

}

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/* Third case: if node to be deleted has two children */If((temp->lchild != NULL) && (temp->rchild != NULL))

{

parent = temp;

temp_succ = temp->rchild;

while(temp_succ != NULL)

{

parent=temp_succ;temp_succ=temp_succ->lchild;

}

temp->data = temp_succ->data;

temp=temp_succ;

}

}

Efficiency of Binary Search Tree

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Efficiency of Binary Search Tree

Operations The timing analysis of searching a binary search

tree, for randomly inserted records is O(log n).

Worst case behavior of a binary search tree

would occur when records are inserted in sorted

order. In this case the search time is O(n), as the

resulting tree consists all NULL left children.

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End of Trees

unit 1 - trees

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